Please don't get put off by the length, all the questions are quite simple, but given the quasi-mathematical context I tried to be precise with the formulation. The more mathematically interesting title question (and the one that's most important for my purposes) is the last one, so if anything please take a look at the end.

Theorem. $ZF$ plus $V=L$ has an $\omega$-model which contains any given countable set of real numbers.

I suspect his terminology might be idiosyncratic, so I'll point out that by an $\omega$-model he means "a model of set theory in which the natural numbers are ordered as they are 'supposed to be'; that is, the sequence of 'natural numbers' of the model is an $\omega$-sequence."

My first (not-so-interesting) question is this:

Is this a proper model-theoretic theorem?

By "proper" here I mean a theorem that is about the properties of a countable set of (first-order) sentences as they are reflected in the properties of their models. That is to say, in the case of Putnam's theorem (which I think is improper), it seems to me that it doesn't really say anything about $ZF$ - instead it describes the metatheory in which it is interpreted using model-theoretic language (and succeeds in doing so by assuming an outer $\omega$ and $\mathbb{P}(\omega)$ which exist independently of the metatheory or the particular set theory being interpreted.)

And the related question:

If I am wrong and it is a proper theorem, then how would one put it in more
contemporary model-theoretic
terminology?

Now to the title question. In his "proof" (quotation marks because I'm not yet sure whether it is a proof or a plausibility argument) of the Theorem, after reducing the statement to something equivalent, he writes:

Now, consider [the sentence '$\mathcal{M}$ is an $\omega$-model for $ZF$ plus $V=L$ and $s$ is represented in $\mathcal{M}$'] in the inner model $V=L$. For every $s$ in the inner model-that is, for every $s \in L$-there is a model-namely $L$ itself-which satisfies "$V=L$" and contains $s$. By the downward L-S Theorem, there is a countable submodel which is elementary equivalent to $L$ and contains $s$.

So here's my question:

Are we allowed to use the downward LST on a proper class-sized inner model such as $L$ in the above case?

In general it seems to me obviously not - any version of the strong LST uses some notion of cardinality which surely cannot apply to $L$. What am I missing here? Putnam does add that strictly speaking the Skolem hull construction is also needed, but I don't see how that would help. Can it?

I will tag this as a reference request too, in case someone knows whether this theorem has been published elsewhere - by Putnam or otherwise. (His footnote says that he proved it in 1963 but provides no more information.)

1 Answer
1

The result you state is not provable in ZFC, and cannot be formalized without going to a stronger theory.

We cannot apply Lowenheim-Skolem (LS) to proper classes; the issue is that we do not have a truth predicate, so the construction of hulls cannot be formalized "from within". Note that partial truth predicates are definable, with truth for $\Sigma_n$ statements being itself $\Sigma_n$, but we cannot define a predicate that works simultaneously for all $n$. I suspect Jech's set theory book addresses this issue. (It is more than a weakness of our approach; Tarski's undefinability of truth theorem shows it is an insurmountable obstacle.) Note that the technical problem disappears when working with set models, since the satisfaction relation becomes $\Delta_1$ definable.

There ought to be an obstacle, since otherwise we could conclude in ZFC that there are set models of ZFC, against the incompleteness theorem.

Assuming stronger axioms, we can formalize LS for certain classes. For example, the existence of $0^\sharp$ is essentially giving us a truth predicate for $L$, and so it allows us to form set sized Skolem hulls of $L$. In fact, $0^\sharp$ gives us many $\alpha$ such that $L_\alpha\prec L$.

Many arguments in set theory take advantage of the reflection theorem, which allow us to find set models of any finite subtheory of ZFC, and so in many arguments we can make do with this weak version of LS. But this does not suffice for Putnam's intended application.

Putnam's notation is standard, though. In an $\omega$-model we always identify standard parts with their transitive version, so in particular, we identify the ${\mathbb N}$ of the model with true $\omega$. Of course, we need this, or it wouldn't make sense to say that a real belongs to the model. Assuming a bit more than ZFC, Putnam's theorem is an easy consequence of Shoenfield's absoluteness.

A good reference for Putnam's proof, the philosophical arguments it generated, and the mathematics behind it, is the PhD thesis of my friend and colleague Johannes Hafner, "A Critical Assessment of Putnam's Model-Theoretic Argument", UC Berkeley, 2006, written under Charles Chihara and Paolo Mancosu.